We should point out that there upper bound on \(\textrm {vb}_n(B_p^m,B_q^m)\)  itself is already proven in
another manuscript of the authors [1]}. However, that
argument goes via the Gaussian measure of the suitable partial colorings. In contrast, here
we give a streamlined argument that shows that the more efficient Lovett-Meka algorithm can be used to
obtain the same bound rather than using convex optimization.
